Remarks on nonlinear uniformly parabolic equations

Indiana University Mathematics Journal

Volumn 47, Issue 4, 1998, Pages 1293-1326

  Crandall, M G ^a   Fok, K ^b   Kocan, M ^c   Swiech A ^d  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Arcadia Design Systems Inc   (United States)

^c AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^d GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000047136     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.1998.47.1561     Document Type: Article

References (34)

1. E. ASTESIANO, Operatori parabolici estremanti, Boll. Un. Mat. Ital. 4 (1968), 403-421.
2. O. V. BESOV, V. P. IL'IN & S. M. NIKOLSKII, Integral Representations of Functions and Imbedding Theorems, Vol. 1, Scripta Series in Mathematics, John Wiley & Sons, 1978.
3. X. CABRÉ, On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48 (1995), 539-570.
4. L. A. CAFFARELLI, Interior a priori estimates for solutions of fully non-linear equations, Ann. Math. 130 (1989), 189-213.
5. L. A. CAFFARELLI & X. CABRÉ, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, 1995.
6. L. CAFFARELLI, M. G. CRANDALL, M. KOCAN & A. ŚWIȨCH, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), 365-397.
7. A. P. CALDERÓN & A. ZYGMUND, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171-226.
8. M. G. CRANDALL, H. ISHII & P. L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67.
9. M. G. CRANDALL, M. KOCAN, P. SORAVIA & A. ŚWIȨCH, On the equivalence of various weak notions of solutions of elliptic PDE's with measurable ingredients, Progress in Elliptic and Parabolic Partial Differential Equations (A. Alvino et al., eds.), Pitman Research Notes in Math., vol. 50, 1996, pp. 136-162.
10. p theory for fully nonlinear parabolic equations, (Preprint).
11. M. G. CRANDALL, M. KOCAN, P. L. LIONS & A. ŚWIȨCH, preprint.
12. p theory for strong solutions of linear parabolic equation with bounded measurable coefficients, Proc. of International Conference on Nonlinear PDE (G. C. Dong & F. H. Lin, eds.), International Academic Press, Beijing, 1992, pp. 46-56.
13. 2,n a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J. 42 (1993), 413-423.
14. L. ESCAURIAZA, A note on Krylov-Tso's parabolic inequality, Proc. Amer. Math. Soc. 115 (1992), 1053-1056.
15. L. C. EVANS, A convergence theorem for solutions of nonlinear second order elliptic equations, Indiana Univ. Math. J. 27 (1978), 875-887.
16. p integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke J. Math. 51 (1984), 997-1016.
17. K. FOK, Some maximum principles and continuity estimates for fully nonlinear elliptic equations of second order, Ph.D. Thesis UCSB.
18. K. FOK, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations 5-6 (1998), 967-983.
19. M. GIAQUINTA, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel Boston Berlin, 1993.
20. D. GILBARG & N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1983.
21. n, Lecture Notes in Math. vol. 481, Springer-Verlag, New York, 1975.
22. R. JENSEN, Uniformly elliptic PDEs with bounded, measurable coefficients, J. Fourier Anal. Appl. 2 (1996), 237-259.
23. N. V. KRYLOV, Sequence of convex functions and estimates of the maximum of the solution of a parabolic equation, English translation in, Siberian Math. J. 17 (1976), 226-236.
24. N. V. KRYLOV, Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel Pub. Co, Dordrecht, 1987.
25. J. M. LASRY & P. L. LIONS, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986), 257-266.
26. G. M. LIEBERMAN, Second Order Parabolic Differential Equations, World Scientific, Singapore New Jersey London Hong Kong, 1996.
27. K. MILLER, Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications, Ann. Mat. Pura Appl. 90 (1971), 297-329.
28. N. S. NADIRASHVILI, Nonclassical solutions to fully nonlinear elliptic equations, preprint.
29. A. I. NAZAROV & N. N. URAL'TSEVA, Convex monotone hulls and estimates of the maximum of the solution of parabolic inequalities, Zap. Nauchn Sem. LOMI 147 (1985), 95-109.
30. S. J. REYE, Harnack inequalities for parabolic equations in general form with bounded measurable coefficients, Research Report R44-84, Centre for Math. Anal., Australian National University.
31. 1,p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations 2 (1997), 1005-1027.
32. K. TSO, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations 10 (1985), 543-553.
33. L. WANG, On the regularity of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math. 45 (1992), 27-76.
34. L. WANG, On the regularity of fully nonlinear parabolic equations: II, Comm. Pure Appl. Math. 45 (1992), 141-178.